A FOOD CHAIN SYSTEM WITH DENSITY-DEPENDENT BIRTH RATE AND IMPULSIVE PERTURBATIONS

2006 ◽  
Vol 09 (03) ◽  
pp. 223-236 ◽  
Author(s):  
SHUWEN ZHANG ◽  
FENYAN WANG ◽  
LANSUN CHEN
Keyword(s):  
Food Chain ◽  
Birth Rate ◽  
Floquet Theory ◽  
Impulsive Differential Equations ◽  
Impulsive Effect ◽  
Top Predator ◽  
Density Dependent ◽  
Chain System ◽  
Predator Eradication ◽  
Fixed Times

We investigate a three-species food chain system with density-dependent birth rate and impulsive effect concerning biological and chemical control strategy — periodic releasing of natural enemies or spraying pesticide at different fixed times. Conditions for the extinction of the prey and top predator are given. By using the Floquet theory of impulsive differential equations and small amplitude perturbation skills, we consider the local stability of the prey and top predator eradication periodic solution. Further, we obtain the conditions of permanence of the system.

scholarly journals Impulsive Perturbations of a Three-Species Food Chain System with the Beddington-DeAngelis Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Younghae Do ◽  
Hunki Baek ◽  
Dongseok Kim
Keyword(s):  
Numerical Simulations ◽  
Functional Response ◽  
Food Chain ◽  
Control Strategy ◽  
Floquet Theory ◽  
Integrated Control ◽  
Comparison Method ◽  
Top Predator ◽  
Chain System ◽  
Theoretical Results

The dynamics of an impulsively controlled three-species food chain system with the Beddington-DeAngelis functional response are investigated using the Floquet theory and a comparison method. In the system, three species are prey, mid-predator, and top-predator. Under an integrated control strategy in sense of biological and chemical controls, the condition for extinction of the prey and the mid-predator is investigated. In addition, the condition for extinction of only the mid-predator is examined. We provide numerical simulations to substantiate the theoretical results.

A FOOD CHAIN SYSTEM WITH HOLLING IV FUNCTIONAL RESPONSES AND IMPULSIVE EFFECT

2008 ◽  
Vol 01 (03) ◽  
pp. 361-375 ◽  
Author(s):  
ZUOLIANG XIONG ◽  
YING XUE ◽  
SHUNYI LI
Keyword(s):  
Periodic Solution ◽  
Food Chain ◽  
Floquet Theory ◽  
Positive Periodic Solution ◽  
Impulsive Effect ◽  
Functional Responses ◽  
Globally Asymptotically Stable ◽  
Chain System ◽  
Maximum Value ◽  
Periodic Constant

In the paper, according to biological and chemical control strategy for pest control, our main purpose is to construct a three trophic level food chain system with Holling IV functional responses and periodic constant impulsive effect concerning integrated pest management (IPM), and investigate the dynamic behaviors of this system. By using the Floquet theory and comparison theorem of impulsive differential equation and analytic method, we prove that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Further, condition for permanence of the system is established. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level.

Stochastic numerical investigations for nonlinear three-species food chain system

2021 ◽  
Author(s):  
Zulqurnain Sabir
Keyword(s):  
Food Chain ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Interior Point Algorithm ◽  
Top Predator ◽  
Chain System ◽  
Comparison Of The Results ◽  
Global And Local ◽  
Algorithm Scheme

In this work, three-dimensional nonlinear food chain system is numerically treated using the computational heuristic framework of artificial neural networks (ANNs) together with the proficiencies of global and local search approaches based on genetic algorithm (GA) and interior-point algorithm scheme (IPAS), i.e. ANN–GA–IPAS. The three-dimensional food chain system consists of prey populations, specialist predator and top-predator. The formulation of an objective function using the differential system of three-species food chain and its initial conditions is presented and the optimization is performed by using the hybrid computing efficiency of GA–IPAS. The achieved numerical solutions through ANN–GA–IPAS to solve the nonlinear three-species food chain system are compared with the Adams method to validate the exactness of the designed ANN–GA–IPAS. The comparison of the results is presented to authenticate the correctness of the designed ANN–GA–IPAS for solving the nonlinear three-species food chain system. Moreover, statistical representations for 40 independent trials and 30 variables validate the efficacy, constancy and reliability of ANN–GA–IPAS.

scholarly journals A Three-Species Food Chain System with Two Types of Functional Responses

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Younghae Do ◽  
Hunki Baek ◽  
Yongdo Lim ◽  
Dongkyu Lim
Keyword(s):  
Food Chain ◽  
Equilibrium Points ◽  
Ecological Systems ◽  
Largest Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Top Predator ◽  
Functional Responses ◽  
Chain System ◽  
Dynamical Behaviors ◽  
Theoretical Results

In recent decades, many researchers have investigated the ecological models with three and more species to understand complex dynamical behaviors of ecological systems in nature. However, when they studied the models with three species, they have just considered the functional responses between prey and mid-predator and between mid-predator and top predator as the same type. However, in the paper, in order to describe more realistic ecological world, a three-species food chain system with two types of functional response, Holling type and Beddington-DeAngelis type, is considered. It is shown that this system is dissipative. Also, the local and global stability of equilibrium points of the system is established. In addition, conditions for the persistence of the system are found according to the existence of limit cycles. Some numerical examples are given to substantiate our theoretical results. Moreover, we provide numerical evidence of the existence of chaotic phenomena by illustrating bifurcation diagrams of system and by calculating the largest Lyapunov exponent.

scholarly journals Dynamics of an Almost Periodic Food Chain System with Impulsive Effects

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yaqin Li ◽  
Wenquan Wu ◽  
Tianwei Zhang
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Food Chain ◽  
Impulsive Differential Equations ◽  
Almost Periodic Solution ◽  
Asymptotically Stable ◽  
Almost Periodic ◽  
Chain System ◽  
Positive Almost Periodic Solution ◽  
Impulsive Perturbations

In order to obtain a more accurate description of the ecological system perturbed by human exploitation activities such as planting and harvesting, we need to consider the impulsive differential equations. Therefore, by applying the comparison theorem and the Lyapunov method of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution in a food chain system with almost periodic impulsive perturbations. The method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations. Finally, an example and numerical simulations are given to illustrate that our results are feasible.

scholarly journals Impulsive Control Strategy for a Nonautonomous Food-Chain System with Multiple Delays

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Baodan Tian ◽  
Yanhong Qiu ◽  
Yucai Ding
Keyword(s):  
Differential Equations ◽  
Food Chain ◽  
Impulsive Control ◽  
Control Strategies ◽  
Functional Differential Equations ◽  
Impulsive Differential Equations ◽  
Effective Control ◽  
Multiple Delays ◽  
Chain System ◽  
Holling Ii Functional Response

A nonautonomous food-chain system with Holling II functional response is studied, in which multiple delays of digestion are also considered. By applying techniques in differential inequalities, comparison theorem in ordinary differential equations, impulsive differential equations, and functional differential equations, some effective control strategies are obtained for the permanence of the system. Furthermore, effects of some important coefficients and delays on the permanence of the system are intuitively and clearly shown by series of numerical examples.

A Stochastic Hierarchical Population System: Excitement, Extinction and Transition to Chaos

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Irina Bashkirtseva ◽  
Tatyana Perevalova ◽  
Lev Ryashko
Keyword(s):  
Mathematical Modeling ◽  
Food Chain ◽  
Analytical Method ◽  
Three Dimensional ◽  
Biological Parameters ◽  
Top Predator ◽  
Population System ◽  
Modeling And Analysis ◽  
Transition To Chaos ◽  
Random Fluctuations

A problem of the mathematical modeling and analysis of noise-induced transformations of complex oscillatory regimes in hierarchical population systems is considered. As a key example, we use a three-dimensional food chain dynamical model of the interacting prey, predator, and top predator. We perform a comparative study of the impacts of random fluctuations on three key biological parameters of prey growth, predator mortality, and the top predator growth. A detailed investigation of the stochastic excitement, noise-induced transition from order to chaos, and various scenarios of extinction is carried out. Constructive abilities of the semi-analytical method of confidence domains in the analysis of the noise-induced extinction are demonstrated.

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